Trigonometry Examples

$\frac{16}{y + 2} - \frac{7 y}{y^{2} - 7}$

Step 1

Find where the expression $\frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$ is undefined.

$y = - \sqrt{7}, y = - 2, y = \sqrt{7}$

Step 2

Since $\frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$ $\to$ $\infty$ as $y$ $\to$ $- \sqrt{7}$ from the left and $\frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$ $\to$ $- \infty$ as $y$ $\to$ $- \sqrt{7}$ from the right, then $y = - \sqrt{7}$ is a vertical asymptote.

$y = - \sqrt{7}$

Step 3

Since $\frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$ $\to$ $- \infty$ as $y$ $\to$ $- 2$ from the left and $\frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$ $\to$ $\infty$ as $y$ $\to$ $- 2$ from the right, then $y = - 2$ is a vertical asymptote.

$y = - 2$

Step 4

Since $\frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$ $\to$ $\infty$ as $y$ $\to$ $\sqrt{7}$ from the left and $\frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$ $\to$ $- \infty$ as $y$ $\to$ $\sqrt{7}$ from the right, then $y = \sqrt{7}$ is a vertical asymptote.

$y = \sqrt{7}$

Step 5

List all of the vertical asymptotes:

$y = - \sqrt{7}, - 2, \sqrt{7}$

Step 6

$x = \frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$ is an equation of a line, which means there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 7

Find the oblique asymptote using polynomial division.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Reorder terms.

$\frac{9 y^{2} - 14 y - 112}{y^{3} + 2 y^{2} - 7 y - 14}$

Step 7.1.2

Factor out the greatest common factor from each group.

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Step 7.1.2.1

Group the first two terms and the last two terms.

$\frac{9 y^{2} - 14 y - 112}{(y^{3} + 2 y^{2}) - 7 y - 14}$

Step 7.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{9 y^{2} - 14 y - 112}{y^{2} (y + 2) - 7 (y + 2)}$

Step 7.1.3

Factor the polynomial by factoring out the greatest common factor, $y + 2$ .

$\frac{9 y^{2} - 14 y - 112}{(y + 2) (y^{2} - 7)}$

Step 7.2

Expand $(y + 2) (y^{2} - 7)$ .

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Step 7.2.1

Apply the distributive property.

$\frac{9 y^{2} - 14 y - 112}{y (y^{2} - 7) + 2 (y^{2} - 7)}$

Step 7.2.2

Apply the distributive property.

$\frac{9 y^{2} - 14 y - 112}{y \cdot y^{2} + y \cdot - 7 + 2 (y^{2} - 7)}$

Step 7.2.3

Apply the distributive property.

$\frac{9 y^{2} - 14 y - 112}{y \cdot y^{2} + y \cdot - 7 + 2 y^{2} + 2 \cdot - 7}$

Step 7.2.4

Reorder $y$ and $- 7$ .

$\frac{9 y^{2} - 14 y - 112}{y \cdot y^{2} - 7 y + 2 y^{2} + 2 \cdot - 7}$

Step 7.2.5

Raise $y$ to the power of $1$ .

$\frac{9 y^{2} - 14 y - 112}{y \cdot y^{2} - 7 y + 2 y^{2} + 2 \cdot - 7}$

Step 7.2.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 y^{2} - 14 y - 112}{y^{1 + 2} - 7 y + 2 y^{2} + 2 \cdot - 7}$

Step 7.2.7

Add $1$ and $2$ .

$\frac{9 y^{2} - 14 y - 112}{y^{3} - 7 y + 2 y^{2} + 2 \cdot - 7}$

Step 7.2.8

Multiply $2$ by $- 7$ .

$\frac{9 y^{2} - 14 y - 112}{y^{3} - 7 y + 2 y^{2} - 14}$

Step 7.2.9

Move $- 7 y$ .

$\frac{9 y^{2} - 14 y - 112}{y^{3} + 2 y^{2} - 7 y - 14}$

Step 7.3

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$y^{3}$

$2 y^{2}$

$7 y$

$14$

$9 y^{2}$

$14 y$

$112$

Step 7.4

The final answer is the quotient plus the remainder over the divisor.

$0 + \frac{y^{3} + 11 y^{2} - 21 y - 126}{y^{3} + 2 y^{2} - 7 y - 14}$

Step 7.5

The oblique asymptote is the polynomial portion of the long division result.

$y = 0$

Step 8

This is the set of all asymptotes.

Vertical Asymptotes: $y = - \sqrt{7}, - 2, \sqrt{7}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 0$

Step 9